The standards for INT vs illegally batted ball are different.
The question of INT/foul ball turns on whether the batter has left his stance as a batter and become a batter runner. If he's still in his batting stance — 2 feet on the ground in the box — and a batted ball hits him, it's a foul ball. If he's begun to run — even if a foot has not landed outside the box — he's going to be called out for INT. This is a violation by a runner (namely, the BR).
To be batted illegally (in the relevant way), the batter's foot must be on the ground entirely outside the box at the moment he contacts the pitch with his bat. The video cannot have been illegally batted in this sense, because it was no longer a pitched ball, but a batted ball. This is a violation by a batter.
"Legal position in the batter's box" = the position used by a batter when batting. Both feet are on the ground in that position, as it is rather uncommon to hit a pitch with one or both feet in the air. Start the swing and stride? Sure. But not when actually offering at the pitch.